Navigable maps of structural brain networks across species

Connectomes are spatially embedded networks whose architecture has been shaped by physical constraints and communication needs throughout evolution. Using a decentralized navigation protocol, we investigate the relationship between the structure of the connec- tomes of different species and their spatial layout. As a navigation strategy, we use greedy routing where nearest neighbors, in terms of geometric distance, are visited. We measure the fraction of successful greedy paths and their length as compared to shortest paths in the topology of connectomes. In Euclidean space, we find a striking difference between the nav- igability properties of mammalian and non-mammalian species, which implies the inability of Euclidean distances to fully explain the structural organization of their connectomes. In con- trast, we find that hyperbolic space, the effective geometry of complex networks, provides almost perfectly navigable maps of connectomes for all species, meaning that hyperbolic distances are exceptionally congruent with the structure of connectomes. Hyperbolic maps therefore offer a quantitative meaningful representation of connectomes that suggests a new cartography of the brain based on the combination of its connectivity with its effective geometry rather than on its anatomy only. Hyperbolic maps also provide a universal frame- work to study decentralized communication processes in connectomes of different species and at different scales on an equal footing

Hyperbolic Geometry of Complex Networks

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as non-interacting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

From Graph Theory to Network Science: The Natural Emergence of Hyperbolicity (Tutorial)

Network science is driven by the question which properties large real-world networks have and how we can exploit them algorithmically. In the past few years, hyperbolic graphs have emerged as a very promising model for scale-free networks. The connection between hyperbolic geometry and complex networks gives insights in both directions: (1) Hyperbolic geometry forms the basis of a natural and explanatory model for real-world networks. Hyperbolic random graphs are obtained by choosing random points in the hyperbolic plane and connecting pairs of points that are geometrically close. The resulting networks share many structural properties for example with online social networks like Facebook or Twitter. They are thus well suited for algorithmic analyses in a more realistic setting. (2) Starting with a real-world network, hyperbolic geometry is well-suited for metric embeddings. The vertices of a network can be mapped to points in this geometry, such that geometric distances are similar to graph distances. Such embeddings have a variety of algorithmic applications ranging from approximations based on efficient geometric algorithms to greedy routing solely using hyperbolic coordinates for navigation decisions

D-Mercator: multidimensional hyperbolic embedding of real networks

One of the pillars of the geometric approach to networks has been the development of model-based mapping tools that embed real networks in its latent geometry. In particular, the tool Mercator embeds networks into the hyperbolic plane. However, some real networks are better described by the multidimensional formulation of the underlying geometric model. Here, we introduce $D$-Mercator, a model-based embedding method that produces multidimensional maps of real networks into the $(D+1)$-hyperbolic space, where the similarity subspace is represented as a $D$-sphere. We used $D$-Mercator to produce multidimensional hyperbolic maps of real networks and estimated their intrinsic dimensionality in terms of navigability and community structure. Multidimensional representations of real networks are instrumental in the identification of factors that determine connectivity and in elucidating fundamental issues that hinge on dimensionality, such as the presence of universality in critical behavior

Scale-free networks and scalable interdomain routing

Trabalho apresentado no âmbito do Mestrado em Engenharia Informática, como requisito parcial para obtenção do grau de Mestre em Engenharia InformáticaThe exponential growth of the Internet, due to its tremendous success, has brought to light some limitations of the current design at the routing and arquitectural level, such as scalability and convergence as well as the lack of support for traffic engineering, mobility, route differentiation and security. Some of these issues arise from the design of the current architecture, while others are caused by the interdomain routing scheme - BGP. Since it would be quite difficult to add support for the aforementioned issues, both in the interdomain architecture and in the in the routing scheme, various researchers believe that a solution can only achieved via a new architecture and (possibly) a new routing scheme. A new routing strategy has emerged from the studies regarding large-scale networks, which is suitable for a special type of large-scale networks which characteristics are independent of network size: scale-free networks. Using the greedy routing strategy a node routes a message to a given destination using only the information regarding the destination and its neighbours, choosing the one which is closest to the destination. This routing strategy ensures the following remarkable properties: routing state in the order of the number of neighbours; no requirements on nodes to exchange messages in order to perform routing; chosen paths are the shortest ones. This dissertation aims at: studying the aforementioned problems, studying the Internet configuration as a scale-free network, and defining a preliminary path onto the definition of a greedy routing scheme for interdomain routing

Greedy routing and virtual coordinates for future networks

At the core of the Internet, routers are continuously struggling with ever-growing routing and forwarding tables. Although hardware advances do accommodate such a growth, we anticipate new requirements e.g. in data-oriented networking where each content piece has to be referenced instead of hosts, such that current approaches relying on global information will not be viable anymore, no matter the hardware progress. In this thesis, we investigate greedy routing methods that can achieve similar routing performance as today but use much less resources and which rely on local information only. To this end, we add specially crafted name spaces to the network in which virtual coordinates represent the addressable entities. Our scheme enables participating routers to make forwarding decisions using only neighbourhood information, as the overarching pseudo-geometric name space structure already organizes and incorporates "vicinity" at a global level. A first challenge to the application of greedy routing on virtual coordinates to future networks is that of "routing dead-ends" that are local minima due to the difficulty of consistent coordinates attribution. In this context, we propose a routing recovery scheme based on a multi-resolution embedding of the network in low-dimensional Euclidean spaces. The recovery is performed by routing greedily on a blurrier view of the network. The different network detail-levels are obtained though the embedding of clustering-levels of the graph. When compared with higher-dimensional embeddings of a given network, our method shows a significant diminution of routing failures for similar header and control-state sizes. A second challenge to the application of virtual coordinates and greedy routing to future networks is the support of "customer-provider" as well as "peering" relationships between participants, resulting in a differentiated services environment. Although an application of greedy routing within such a setting would combine two very common fields of today's networking literature, such a scenario has, surprisingly, not been studied so far. In this context we propose two approaches to address this scenario. In a first approach we implement a path-vector protocol similar to that of BGP on top of a greedy embedding of the network. This allows each node to build a spatial map associated with each of its neighbours indicating the accessible regions. Routing is then performed through the use of a decision-tree classifier taking the destination coordinates as input. When applied on a real-world dataset (the CAIDA 2004 AS graph) we demonstrate an up to 40% compression ratio of the routing control information at the network's core as well as a computationally efficient decision process comparable to methods such as binary trees and tries. In a second approach, we take inspiration from consensus-finding in social sciences and transform the three-dimensional distance data structure (where the third dimension encodes the service differentiation) into a two-dimensional matrix on which classical embedding tools can be used. This transformation is achieved by agreeing on a set of constraints on the inter-node distances guaranteeing an administratively-correct greedy routing. The computed distances are also enhanced to encode multipath support. We demonstrate a good greedy routing performance as well as an above 90% satisfaction of multipath constraints when relying on the non-embedded obtained distances on synthetic datasets. As various embeddings of the consensus distances do not fully exploit their multipath potential, the use of compression techniques such as transform coding to approximate the obtained distance allows for better routing performances

Dynamics of spectral algorithms for distributed routing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 109-117).In the past few decades distributed systems have evolved from man-made machines to organically changing social, economic and protein networks. This transition has been overwhelming in many ways at once. Dynamic, heterogeneous, irregular topologies have taken the place of static, homogeneous, regular ones. Asynchronous, ad hoc peer-to-peer networks have replaced carefully engineered super-computers, governed by globally synchronized clocks. Modern network scales have demanded distributed data structures in place of traditionally centralized ones. While the core problems of routing remain mostly unchanged, the sweeping changes of the computing environment invoke an altogether new science of algorithmic and analytic techniques. It is these techniques that are the focus of the present work. We address the re-design of routing algorithms in three classical domains: multi-commodity routing, broadcast routing and all-pairs route representation. Beyond their practical value, our results make pleasing contributions to Mathematics and Theoretical Computer Science. We exploit surprising connections to NP-hard approximation, and we introduce new techniques in metric embeddings and spectral graph theory. The distributed computability of "oblivious routes", a core combinatorial property of every graph and a key ingredient in route engineering, opens interesting questions in the natural and experimental sciences as well. Oblivious routes are "universal" communication pathways in networks which are essentially unique. They are magically robust as their quality degrades smoothly and gracefully with changes in topology or blemishes in the computational processes. While we have only recently learned how to find them algorithmically, their power begs the question whether naturally occurring networks from Biology to Sociology to Economics have their own mechanisms of finding and utilizing these pathways. Our discoveries constitute a significant progress towards the design of a self-organizing Internet, whose infrastructure is fueled entirely by its participants on an equal citizen basis. This grand engineering challenge is believed to be a potential technological solution to a long line of pressing social and human rights issues in the digital age. Some prominent examples include non-censorship, fair bandwidth allocation, privacy and ownership of social data, the right to copy information, non-discrimination based on identity, and many others.by Petar Maymounkov.Ph.D

Measuring Effectiveness of Address Schemes for AS-level Graphs

This dissertation presents measures of efficiency and locality for Internet addressing schemes. Historically speaking, many issues, faced by the Internet, have been solved just in time, to make the Internet just work~\cite{justWork}. Consensus, however, has been reached that today\u27s Internet routing and addressing system is facing serious scaling problems: multi-homing which causes finer granularity of routing policies and finer control to realize various traffic engineering requirements, an increased demand for provider-independent prefix allocations which injects unaggregatable prefixes into the Default Free Zone (DFZ) routing table, and ever-increasing Internet user population and mobile edge devices. As a result, the DFZ routing table is again growing at an exponential rate. Hierarchical, topology-based addressing has long been considered crucial to routing and forwarding scalability. Recently, however, a number of research efforts are considering alternatives to this traditional approach. With the goal of informing such research, we investigated the efficiency of address assignment in the existing (IPv4) Internet. In particular, we ask the question: ``how can we measure the locality of an address scheme given an input AS-level graph?\u27\u27 To do so, we first define a notion of efficiency or locality based on the average number of bit-hops required to advertize all prefixes in the Internet. In order to quantify how far from ``optimal the current Internet is, we assign prefixes to ASes ``from scratch in a manner that preserves observed semantics, using three increasingly strict definitions of equivalence. Next we propose another metric that in some sense quantifies the ``efficiency of the labeling and is independent of forwarding/routing mechanisms. We validate the effectiveness of the metric by applying it to a series of address schemes with increasing randomness given an input AS-level graph. After that we apply the metric to the current Internet address scheme across years and compare the results with those of compact routing schemes